Supercurrents oscillate with quantum interference in nanojunctions

Mikhail Kalenkov of the I.E.Tamm Department of Theoretical Physics, P.N.Lebedev Physical Institute, and Andrei Zaikin of the same institute, alongside National Research University Higher School of Economics, have demonstrated that superconducting nanojunctions with barrier transmissions slightly below unity exhibit pronounced, coherent oscillations of the supercurrent as a function of the Josephson phase. The researchers derived an effective Hamiltonian to model quantum dynamics, then solved a Schrödinger-like equation to obtain wave functions for Andreev levels and evaluate electric current. This work expands understanding beyond full-transmission junctions, offering insights into systems with diffusive barriers and refining existing models of quantum current flow.

Andreev States and Effective Hamiltonian Derivation

Researchers Mikhail S. Zaikin of the I.E.Tamm Department of Theoretical Physics, P.N.Lebedev Physical Institute, and National Research University Higher School of Economics, developed this Hamiltonian, which allows for the derivation of a Schrödinger-like equation. This equation reveals the wave functions that describe the behavior of these levels and enables the calculation of electric current flowing through the junction under an applied voltage.

The theoretical framework builds upon existing understanding of both full and arbitrary transmissions in superconducting weak links, but specifically addresses a gap in knowledge concerning junctions with barriers just below unity. The researchers formulated a Schrödinger-like equation by substituting into their derived Hamiltonian, yielding a Hermitian effective Hamiltonian, H, with linear terms in χ(t) retained to account for long-term evolution and prevent decoherence.

This approach is complementary to, and fully consistent with, the established physical picture of multiple Andreev reflections. The resulting equation allows for the construction of solutions describing the entire phase interval, even in regions where adiabatic approximations break down due to strong coupling between Andreev levels. Crucially, the team’s model accounts for the non-unitary time evolution of Andreev states, a factor often overlooked in simpler treatments.

Standard quantum mechanical approaches fail to adequately describe these states, which represent a superposition of quasiparticles and holes, especially when subjected to a voltage bias that introduces both dissipation and decoherence. By solving the Schrödinger-like equation, the researchers obtained an expression for the current-phase relation, revealing that quantum interference between Andreev states can induce coherent oscillations in the supercurrent. This interference is particularly pronounced when the barrier transmission is low, but not zero, and the behavior of the system is notable near pi.

The analysis involved deriving an integral kernel for the inverse operator (W + a)^(-1), expressed in terms of the scattering matrix elements, d(t) and r(t). The researchers found that the behavior of the system closely resembles that of Landau-Zener tunneling, a quantum mechanical phenomenon where a system transitions between states due to a slowly varying external field. The researchers state that they derived equations for d and delta, but do not provide the full formulas.

This parameter, denoted as ‘s’ in their calculations, is related to the adiabadicity parameter. The researchers state, “We demonstrate that at low but non-zero values of R, CPR of a voltage-biased superconducting junction may exhibit pronounced coherent oscillations caused by quantum interference between Andreev states.” These oscillations are expected to be particularly significant in junctions with diffusive barriers, suggesting potential implications for the design and performance of future superconducting devices. The team’s work provides a refined theoretical foundation for understanding current flow in these complex nanoscale systems, opening avenues for further exploration of quantum phenomena in superconductivity.

Landau-Zener Tunneling in Superconducting Nanojunctions

Superconducting nanojunctions, increasingly vital components in advanced quantum circuits, exhibit coherent oscillations in supercurrent flow when barrier transmissions fall just below unity, according to new theoretical work by Mikhail S. Kalenkov and Andrei Zaikin. These oscillations, stemming from quantum interference between Andreev states, demonstrate qualitatively new features as compared to the standard tunneling limit, and refine existing models of these nanoscale devices. The research details a microscopic theory and effective Hamiltonian to model the quantum dynamics within these junctions, offering a more accurate prediction of current flow under voltage bias.

This approach allows for a more complete description of the quantum behavior within the junction, considering Andreev states as non-trivial superpositions of quasiparticles and holes, and the non-unitary time evolution induced by the voltage bias. This behavior arises from the quantum interference of Andreev states, a phenomenon amplified by the specific transmission characteristics of the junction.

The quantitative relationship between the barrier characteristics and the oscillations is described by a parameter s, related to the adiabadicity parameter. The team derived equations for the system, including d = -e^(-π s), and the phase shift, δ, is given by -π/4 + s lns/e- argΓ(is), with Γ(x) being the Euler gamma-function. This finding suggests a relationship between the barrier’s properties and the coherence of the supercurrent. The implications of this research extend beyond fundamental physics, offering a pathway toward improved performance in areas such as sensitive detectors and quantum information processing.

Quasiclassical Eilenberger-Keldysh Equations for Current Calculation

This refinement of existing models addresses a gap in understanding previously focused on either full or arbitrary transmission scenarios. The researchers employed a microscopic theory to derive an effective Hamiltonian, a mathematical tool used to model the quantum dynamics of Andreev states within the nanojunctions. The resulting model predicts that the current-phase relation, the relationship between the voltage applied to the junction and the resulting current, can exhibit pronounced coherent oscillations.

The researchers detail how, near the point where the phase reaches pi, the solution to the Schrödinger-like equation reduces to a form standard for Landau-Zener tunneling, allowing them to determine the scattering matrix and ultimately, the supercurrent. By understanding and controlling these coherent oscillations, it may be possible to engineer nanojunctions with enhanced sensitivity or tailored current-phase relationships for specific applications.

Coherent Oscillations of Supercurrent with Voltage Bias

The team’s work moves beyond earlier models that focused on either complete or arbitrary transmission rates, refining understanding of the intermediate range where barrier transmission is near, but not equal to, one. The resulting current-phase relation (CPR) exhibits these oscillations, consistent with well-understood behavior in fully transmitting junctions.

This behavior is particularly pronounced in junctions with diffusive barriers, suggesting that these materials could be engineered to enhance the effect. The researchers detail how the integral kernel for the inverse operator within their model mathematically describes this phenomenon, providing a precise means of predicting and controlling the oscillations.

The team’s analysis builds on previous work by Averin and Bardas, who attempted to answer a question using a physical picture of Landau-Zener tunneling between Andreev levels. However, the current study addresses factors important to consider in that earlier model, including the treatment of Andreev states as non-trivial superpositions of quasiparticles and holes, and the non-unitary time evolution induced by the voltage bias. The researchers employed a formalism that allows for a consistent treatment of both dissipation and decoherence, factors often neglected in simpler models.

By carefully constructing the scattering matrix, they were able to accurately describe the evolution of Andreev states and predict the resulting supercurrent. This detailed analysis, combined with the derived effective Hamiltonian, provides a powerful tool for understanding and manipulating quantum phenomena in superconducting nanojunctions.

Josephson Phase Dependence and Andreev Level Energies

Superconducting nanojunctions, crucial components in emerging quantum technologies, don’t always behave as predicted by conventional models. While established theory adequately describes junctions with near-perfect transmission of electrons, a new analysis reveals a surprising phenomenon in those with slightly reduced transmission: pronounced oscillations in the supercurrent as the Josephson phase changes.

By solving this equation, they were able to accurately describe how the Andreev states evolve under an applied voltage, revealing the origins of the observed supercurrent oscillations. The current study addresses factors important to consider in that earlier model, particularly the treatment of Andreev states as non-trivial superpositions of quasiparticles and holes, and the non-unitary time evolution induced by the voltage bias.

The behavior is notable as the Josephson phase approaches pi. Here, the solution to the Schrödinger-like equation exhibits significant changes. The team’s work extends beyond simply observing these oscillations; it provides a detailed theoretical framework for understanding their origin. They found that the parameter ‘s’, related to the adiabadicity parameter, plays a critical role. The model considers a single channel superconducting junction with transmission D = 1-R, biased by an external voltage V(t).

The researchers arrived at a formally exact expression for electric current, utilizing the Eilenberger-Keldysh equations and Zaitsev boundary conditions. This approach allowed them to accurately evaluate the current across the junction and demonstrate the quantum interference of Andreev states, ultimately revealing the source of the observed supercurrent oscillations.

👉 More information

🗞 Landau-Zener Tunneling and Quantum Interference of Andreev States

✍️ Mikhail S. Kalenkov and Andrei D. Zaikin

🧠 DOI: http://link.aps.org/doi/10.1103/wywl-hk6d